Isometric $F$-spaces of $log$-integrable function

Let $({\mathrm{\Omega }}_i,{\mathcal{A}}_i,{\mu }_i)$ be a measure space with finite measure ${\mu }_i$, and let $(L_{\log}(\Omega_i, \mathcal A_i,\mu_i), |\cdot|{\log,\mu_i})$bea$F$-spaceofall$\log$-integrablefunctionson$(\Omega_i,\mathcal A_i,\mu_1), \ i =1, 2 $\mathrm{.}Itisprovedthat$F$-spaces$(L{\log}(\Omega_1, \mathcal A_1,\mu_1), |\cdot|{\log,\mu_1})\and \(L{\log}(\Omega_2, \mathcal A_2,\mu_2), |\cdot|_{\log,\mu_2})$ are isometric if and only if there exists a measure preserving isomorphism from $({\mathrm{\Omega }}_1,{\mathcal{A}}_1,{\mu }_1)$ onto $({\mathrm{\Omega }}_2,{\mathcal{A}}_2,{\mu }_2)$.